Far-field heat transfer and monochromatic thermal currents in a cylindrical nonreciprocal cavity

TL;DR

This work tackles how nonreciprocal radiative emission and absorption affect far-field heat transfer in a cylindrical cavity. It uses a discretized, specular ray-tracing approach to compute the monochromatic transmission S_{j→i}(ω) between azimuthal elements and to derive the heat-rectification measures H_{ij}(ω) and the many-body nonreciprocity factor ζ, with a generalized Kirchhoff constraint ensuring no persistent equilibrium currents. Key findings show that equilibrium can exhibit nonzero pairwise rectification in nonreciprocal configurations, yet internal currents vanish due to energy balance; under nonequilibrium conditions, nonreciprocity produces tunable rotational heat fluxes, with ideal on–off emitters delivering strong rectification (ζ_8 ≈ 0.85) and Weyl semimetals yielding smaller effects (ζ ≈ 0.061, H_{ij} ≈ 0.028). These results offer design principles for nonreciprocal photonic devices and thermal management, and point to extensions to more complex geometries and dynamic or conductive effects.

Abstract

Breaking Kirchhoff's law of thermal radiation yields new opportunities in one-way radiative thermal transport and circuitry. We investigate its consequences in the far-field regime in cylindrical cavities, by employing a specular ray-tracing algorithm. At thermal equilibrium, we show that violation of Kirchhoff's law yields non-vanishing heat rectification coefficients within different sections of the cavity, which can be tuned for perfect rectification and circulation, while internal monochromatic currents vanish due to the intrinsic coupling between emission and absorption at specular surfaces. This constraint is lifted under nonequilibrium conditions, where rotational heat fluxes within the cavity can be precisely controlled by appropriately combining reciprocal and nonreciprocal materials. These findings open new avenues for thermal management and provide design principles for nonreciprocal photonic devices.

Figures (5)

• Figure 1: Representation of the geometry under consideration. The cylindrical surface is partitioned into $n = 8$ evenly spaced vertical elements, assumed infinite along the $z$-direction. Each element $\Omega_i$, for $i=1, \dots, n$, is defined in Eq. \ref{['eq cylinder partition']} and assigned either a reciprocal or nonreciprocal material exhibiting specular reflection. At each surface point $\mathbf{r}$, emission and absorption obey the generalized Kirchhoff's law in Eq. \ref{['eq:specular_emission']}, and are specified for every direction $\hat{\mathbf{s}}$ by the polar and azimuthal angles $(\theta, \varphi)$, measured relative to the local surface normal $\hat{\mathbf{n}}$. The horizontal angle $\beta \in (-\pi/2, \pi/2)$ characterizes the projection of $\hat{\mathbf{s}}$ on to the xy-plane with respect to $\hat{\mathbf{n}}$.
• Figure 2: Schematic representation of a directed graph illustrating the monochromatic heat exchange in a three element partition of the cylinder at thermal equilibrium. Each node $i$ corresponds to an element $\Omega_i$, while arrows indicate the directional imbalance in the heat rectification coefficient $H_{ij}$. For a reciprocal material configuration, symmetry in the emissivity profile with respect to $\theta$ ensures $S_{i \to j} = S_{j \to i}$, leading to $H_{ij} = 0$, and thus balanced radiative exchange. By contrast, nonreciprocal materials with asymmetric emissivity profiles yield $S_{i \to j} \neq S_{j \to i}$, resulting in $H_{ij} \neq 0$, i.e, a nonzero heat rectification coefficient. For the idealized on-off material, $|H_{ij}| = 0.26$, whereas for a Weyl semimetal it is reduced to $|H_{ij}| = 0.028$ (see Appendix \ref{['sec: weyl semimetal']}).
• Figure 3: Graph representations of the heat rectification coefficient, defined as $H_{i j} = S_{i \to j} - S_{j \to i}$, for a cylindrical configuration partitioned into eight elements. Arrows indicate the direction and magnitude of net radiative exchange between element pairs; absence of an arrow implies $H_{i j} = 0$. (a) A reciprocal material with a symmetric emission and absorption profile is considered. Due to reciprocity and angular symmetry, the heat rectification coefficients vanish for all element pairs, i.e., $H_{i j} = 0$, consistent with detailed balance and reciprocity constraints. (b) A nonreciprocal idealized on–off emitter is used, characterized by high emissivity into the right hemisphere ($\beta>0$) and high absorptivity from the left ($\beta<0$). This asymmetry leads to a pronounced directional transmission pattern, with significant nonzero values of $H_{i j}$, forming circular heat channels. The configuration exhibits $C_8$ rotational symmetry and enhances directional heat rectification coefficients by nearly an order of magnitude compared to realistic nonreciprocal materials. (c) A mixed configuration where odd-numbered elements in green are nonreciprocal (on–off type), and even-numbered elements in blue are reciprocal blackbodies. Despite the presence of only four nonreciprocal elements, nonzero heat rectification coefficients emerge across nearly all element pairs due to indirect radiative coupling mediated by reflection and emission. In all cases, the total net power exchange at each element vanishes, as required by the second law of thermodynamics.
• Figure 4: Monochromatic heat flux vector fields in an eight-element cylindrical configuration under different thermal and material conditions. (a) At thermal equilibrium, with all elements composed of reciprocal blackbody material, no net thermal currents are observed, either in the interior or at the boundary. (b) At thermal equilibrium, with a nonreciprocal on-off elements (green), the internal thermal currents vanish, but a nonzero boundary inflow monochromatic heat appears, reflecting a directional imbalance in surface absorption and emission due to the asymmetric emissivity profile. (c) Out of equilibrium, with reciprocal blackbody elements at even indices (light blue, cold) and odd indices (blue, hot), the heat-flux vectors point symmetrically from hot to cold elements, yielding no preferred rotational direction. (d) In the same nonequilibrium configuration, but with nonreciprocal elements (green, hot), the heat flux vectors form a counterclockwise rotational pattern, driven by an asymmetric emissivity that favors emission into the right hemisphere.
• Figure 5: Monochromatic heat flux vector field in an eight-element cylindrical configuration using an abrupt, nonphysical emissivity profile defined by $\epsilon(\beta) = 1$ for $\beta \geq 0$ and $\epsilon(\beta) = 0$ for $\beta < 0$. This artificial discontinuity leads to unbalanced emission and absorption: rays are strongly emitted in the right hemisphere but cannot be absorbed from the same directions. As a result, the simulation produces unrealistic internal thermal currents that violate detailed balance.